Techniques of Integration

Below you will find the exercises we suggest you work on in connection with the Techniques of Integration theme and the project problems which are to be done for the exercise classes during October 21-25.

Section 6.1: Integration by Parts
6.1.1, 6.1.4, 6.1.13, 6.1.33, 6.1.37.

Relevant Maple Worksheet: -:Partial integration

Section 6.2: Integrals of Rational Functions
6.2.1, 6.2.3, 6.2.7, 6.2.9, 6.2.20, 6.2.27.

Relevant Maple worksheet: -:Rational Functions

Section 6.3: Inverse Substitutions
6.3.1, 6.3.8, 6.3.43, 6.3.46, 6.3.49, 6.3.51.

Relevant Maple worksheet: -:Substitutions

Section 6.4: Other Methods for Evaluating Integrals
6.4.1, 6.4.3.

Relevant pencast: The method of undetermined coefficients

Section 6.5: Improper Integrals
6.5.1, 6.5.3, 6.5.5, 6.5.6, 6.5.27, 6.5.31.

Section 6.6: The Trapezoid and Midpoint Rules
6.6.1, 6.6.5.

Section 6.7: Simpson's Rule
6.7.1, 6.7.5.

Relevant Maple worksheet: -:Numerical Integration

Section 6.8: Other Aspects of Approximate Integration
6.8.1, 6.8.7, 6.8.9, 6.8.10.

These problems are to be presented during October 21-25. See here for where you should meet, and here to sign up for presenting a problem.

PDF-version of the exercises
Problem 1 (Maple TA)

Find the smallest positive number \(x\) such that \(F(x) = \frac{1}{20 \pi}\) where \(F\) is given by \[F(x) = \int_0^x e^{5\sqrt{3}\pi t}\sin(15\pi t)\,dt.\]

Your answer should be an exact positive rational number.

Hint: 1) Try first to find an antiderivative, \(G(t)\), of the integrand by using integration by parts twice.

2) Then \(F(x) = G(x) - G(0).\)

3) You may find it useful to remember that \(\tan\pi/3 = \sqrt{3}.\)

Problem 2 (Exam 1996 in 75011, problem 2)

a) Calculate the indefinite integral \[\int \frac{2}{(x+1)(x^2+1)} \ dx.\] Hint: :Rational Functions and Problem 6.2.20.

b) Find the exact value of the improper integral \[\int\limits_{1}^{\infty} \frac{2}{(x+1)(x^2+1)} \ dx.\]

Hint: Another example of an improper integral

Problem 3 (Exam in MAT1001, UiO)

Find \(a>0\) so that the integral \[\int\limits_{0}^{\infty}\cos x \mathrm{e}^{-ax} \ dx \] has maximal value. Find this value. Hint: :.Partial integration, and example 4 page 335 in Adams.

Problem 4 (Coulomb's Law)

Two electrically charged particles repel each other if they have the same charge, and attract each other if they have opposite charge. According to Coulomb's law the force of attraction/repulsion is given by \[F=k\frac{q_1q_2}{r^2}\] where \(k\) is a constant, \(q_1\) and \(q_2\) is the charge of the particles and \(r\) is the distance between them. The work required to move one particle from a distance \(a\) to a distance \(b\) away from the other is given by \[W=-\int\limits_{a}^{b}F(r) \ dr.\]

a) Assume that the particles are at a distance \(d\) from each other, and have opposite charge. How much work is required to move one particle infinitely far from the other?

b) Assume again that the particles are at a distance \(d\) from each other, and have the same charge. How much work is required to move the particles together?